Probability tells you how likely an event is. It lives on a simple scale from 0% (the event never happens) to 100% (the event always happens). In everyday decisions — from sports betting to medical tests to rolling dice — the questions are almost always the same: What are the chances of A? What are the chances of A and B both happening? What are the chances of at least one of them happening? This probability calculator answers all four of those questions in one step, using the classic probability laws you learned in introductory statistics.
Type the number of trials and the number of successes for event A and event B, choose whether the events are independent or mutually exclusive, and every result updates instantly.
How to Use
- Enter the Event A — Trials — the total number of opportunities you gave event A. Flipping a coin 10 times, rolling a die 50 times, or surveying 1,000 people are all examples.
- Enter the Event A — Successes — how many of those trials actually produced the outcome you care about.
- Repeat the two fields for Event B.
- Choose the Relationship Between A and B. Pick Independent when the two events do not influence each other (one die roll, one coin toss). Pick Mutually Exclusive when the two events cannot happen at the same time (drawing a heart vs. a spade from one card, or rolling a 2 vs. a 5 on one die).
- Read the result card: P(A), P(not A), P(A and B), and P(A or B).
All four numbers are shown as percentages, and they always add up in a way that makes intuitive sense.
The Four Core Probability Laws
Probability is built on a small set of rules. Once you know them, you can check the calculator's output by hand.
1. Basic probability. The chance of an event is its successes divided by its trials:
P(A) = successesA / trialsA
Flipping heads on 3 out of 10 coin tosses gives P(A) = 3 / 10 = 30%.
2. The complement rule. The chance that A does not happen is everything left over:
P(not A) = 1 − P(A)
If P(A) = 30%, then P(not A) = 70%. The two always sum to 100%.
3. Independent events (multiplication rule). When A and B do not affect each other, the chance of both happening together is the product:
P(A and B) = P(A) × P(B)
If a 30% event and a 20% event are independent, both happen 30% × 20% = 6% of the time.
4. The addition rule. The chance that either A or B happens depends on their relationship:
- Independent: P(A or B) = P(A) + P(B) − P(A and B). We subtract the overlap so we do not double-count the case where both happen.
- Mutually exclusive: P(A and B) = 0 by definition, so the overlap vanishes and P(A or B) = P(A) + P(B).
A worked example: P(A) = 30%, P(B) = 20%, independent. P(A and B) = 6%, and P(A or B) = 30% + 20% − 6% = 44%. If the same two events were mutually exclusive, P(A and B) would be 0% and P(A or B) would be 50%.
Independent vs. Mutually Exclusive: Which Is It?
People confuse these two ideas constantly, but they are opposite cases. Two events are independent when knowing one happened tells you nothing about the other — flipping heads does not change the next flip. They are mutually exclusive when they physically cannot happen together — a single die cannot show both 2 and 5 at once. A quick test: if P(A and B) = 0, they are mutually exclusive; if P(A and B) = P(A) × P(B), they are independent. They can never be both at the same time (unless one of them is impossible).
Choosing the wrong relationship flips your "or" answer by the size of the overlap, so pause for a second and ask whether the two outcomes could occur on the same trial.
Examples
Example 1 — Two fair coins. You flip one coin 10 times and see heads 3 times (P(A) = 30%). You flip a second coin 10 times and see heads 2 times (P(B) = 20%). The flips are independent, so P(A and B) = 30% × 20% = 6%, and P(A or B) = 30% + 20% − 6% = 44%. P(not A) is the chance of tails on the first coin, 70%.
Example 2 — Drawing from a deck. Event A is "draw a heart" (13 of 52 cards, P(A) = 25%). Event B is "draw a spade" (13 of 52, P(B) = 25%). A single card cannot be both a heart and a spade, so the events are mutually exclusive. P(A and B) = 0%, and P(A or B) = 25% + 25% = 50% — exactly the chance of drawing a red-or-black suit card from the major suits.
Example 3 — A large survey. A company surveys 1,000,000 customers: 250,000 use the mobile app (P(A) = 25%) and 100,000 use the website at least weekly (P(B) = 10%). The two behaviors are treated as independent, so the share who use both is 25% × 10% = 2.5%, and the share who use at least one channel is 25% + 10% − 2.5% = 32.5%. Running a million-row dataset in your head is exactly why a calculator helps.
FAQ
What does P(A) mean?
P(A) is the probability of event A, a number from 0 to 1 (or 0% to 100%). You calculate it by dividing the number of times A occurred by the total number of trials: P(A) = successes / trials. A P(A) of 0 means A is impossible; 1 means A is certain.
How do I calculate P(not A)?
Use the complement rule: P(not A) = 1 − P(A). If P(A) is 0.30 (30%), then P(not A) is 0.70 (70%). The two probabilities always add to 1, because either A happens or it does not.
What is the probability of A and B happening together?
If A and B are independent, multiply their probabilities: P(A and B) = P(A) × P(B). If they are mutually exclusive, P(A and B) = 0, because they cannot happen on the same trial.
What is the probability of A or B happening?
For independent events: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, P(A and B) is 0, so this simplifies to P(A or B) = P(A) + P(B). Subtracting the overlap prevents double-counting trials where both events occur.
What is the difference between independent and mutually exclusive events?
Independent events do not influence each other — one outcome gives you no information about the other. Mutually exclusive events cannot occur together at all. Flipping two coins is independent; drawing one card that is both a heart and a spade is mutually exclusive. Choosing the wrong option changes your "or" result by the size of the overlap.
Can probability be more than 100%?
No. A probability always lives between 0 and 1 (0% and 100%). If your calculation gives a value above 1 or below 0, you likely forgot to divide successes by trials, or you double-counted an overlap. The complement rule and the addition rule are the fastest ways to sanity-check a result.
Disclaimer
This probability calculator is provided for informational purposes only and does not constitute financial, medical, legal, or other professional advice. Results are based on the inputs you enter and the standard probability laws; real-world events may be correlated, biased, or sampled too thinly to generalize. Always consult a qualified professional before making important decisions based on calculated probabilities.